Proper divisors

Proper divisors is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

The proper divisors of a positive integer N are those numbers, other than N itself, that divide N without remainder, and always include 1.

For example the proper divisors of 6 are 1, 2, and 3. The proper divisors of 100 are 1, 2, 4, 5, 10, 20, 25, and 50.

Task

1. Create a routine to generate all the proper divisors of a number.
2. use it to show the proper divisors of the numbers 1 to 10 inclusive.
3. Find a number in the range 1 to 20,000 with the most proper divisors. Show the number and just the count of how many proper divisors it has.

Show all output here.

Java

Works with: Java version 1.5+

<lang java5>import java.util.Collections; import java.util.LinkedList; import java.util.List;

public class Proper{

   public static List<Integer> properDivs(int n){
       List<Integer> divs = new LinkedList<Integer>();
       divs.add(1);
       for(int x = 2; x < n; x++){
           if(n % x == 0) divs.add(x);
       }
       
       Collections.sort(divs);
       
       return divs;
   }
   
   public static void main(String[] args){
       for(int x = 1; x <= 10; x++){
           System.out.println(x + ": " + properDivs(x));
       }
       
       int x = 0, count = 0;
       for(int n = 1; n <= 20000; n++){
           if(properDivs(n).size() > count){
               x = n;
               count = properDivs(n).size();
           }
       }
       System.out.println(x + ": " + count);
   }

}</lang>

Output:

1: [1]
2: [1]
3: [1]
4: [1, 2]
5: [1]
6: [1, 2, 3]
7: [1]
8: [1, 2, 4]
9: [1, 3]
10: [1, 2, 5]
15120: 79

Python

A very literal interpretation <lang python>>>> def proper_divs2(n): ... return {x for x in range(1, (n + 1) // 2 + 1) if n % x == 0} ... >>> [proper_divs2(n) for n in range(1, 11)] [{1}, {1}, {1}, {1, 2}, {1}, {1, 2, 3}, {1}, {1, 2, 4}, {1, 3}, {1, 2, 5}] >>> >>> n, length = max(((n, len(proper_divs2(n))) for n in range(1, 20001)), key=lambda pd: pd[1]) >>> n 15120 >>> length 79 >>> </lang>